> ...he's quoting what someone described as a "fundamental tenet of numerical linear algebra" as "anything Jordan decomposition can do, Schur can do better"...

I suspect a reason to keep this little slogan in mind is that the Jordan decomposition/Jordan canonical form is taught as a tool in various classroom settings. I first learned it in studying linear feedback systems of the form:

    x(t+1) = A x(t) + u(t)

where u(t) is a system input. These are used in control systems, audio filters, etc.

In this case, the Jordan canonical form of the nXn matrix A has many properties that explain intuitively what the system is doing.

For instance, if there are two sub-blocks in the Jordan decomposition, these can be interpreted as two smaller non-interacting feedback systems. So in that case the original nXn system is equivalent to two smaller systems.

Anyway, because the Jordan canonical form is remembered as a relevant toolbox item, sometimes people try to use it for numerical problems as well -- e.g., when the "A" above is estimated from data.

And it doesn't work well for that purpose because the decomposition is very unstable and therefore the above interpretations are similarly shaky.